Finite Group Databasebeta

← C7×#24326 #13201542

Subgroups — C7×#24326 #13201542

Order 1792.

1330 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×D4×Q8C211yesmaximal, characteristic, complemented
2C2×C7×#474C211yesmaximal, characteristic, complemented
3C2×C4×C7×D8C211yesmaximal, characteristic
4C7×#2538C211yesmaximal, characteristic, complemented
5C7×#2057C212yesmaximal, complemented
6C7×#2023C212yesmaximal, complemented
7C7×#1990C211yesmaximal, characteristic, complemented
8C7×#1984C211yesmaximal, characteristic, complemented
9C7×#1879C212yesmaximal
10C7×#1862C212yesmaximal, complemented
11C7×#1802C211yesmaximal, characteristic, complemented
12C23×C7×Q8—21no—
13C22×C7×D8C2211yescharacteristic
14C7×D4×Q8—24no—
15C7×D4×Q8—24no—
16C2×C7×C4⋊Q8C2212yes—
17C2×C7×C4⋊Q8C2211yescharacteristic
18C2×C7×C22⋊Q8—21no—
19C2×C7×C22⋊Q8—21no—
20C2×C7×C22⋊Q8—22no—
21C2×C4×C7×Q8C2211yescharacteristic
22C2×C4×C7×D4C2211yescharacteristic
23C2×C4×C7×D4C2212yes—
24C2×C4×C7×D4C2211yescharacteristic
25C7×#474—24no—
26C4×C7×D8—24no—
27C2×C7×#107C2211yescharacteristic
28C2×C7×#107C2212yes—
29C2×C7×C4⋊C8C2211yescharacteristic
30C2×C7×#102C2211yescharacteristic
31C2×C7×#102C2211yescharacteristic
32C2×C7×#102C2212yes—
33C2×C7×#102C2212yes—
34C2×C4×C7×C8C2211yescharacteristic
35C7×#396—22no—
36C7×#390C2212yes—
37C7×#389C2212yes—
38C7×#384C2211yescharacteristic
39C7×#380—21no—
40C7×#340C2212yes—
41C7×#340C2212yes—
42C7×#340C2212yes—
43C7×#336C2212yes—
44C7×#336C2212yes—
45#24326C711yesmaximal, characteristic, direct product factor, Sylow
46C22×C7×Q8—28no—
47C22×C7×Q8—22no—
48C22×C7×Q8—21no—
49C22×C7×Q8—44no—
50C22×C7×Q8—44no—
51C22×C7×Q8D411yescharacteristic
52C22×C7×D4C2311yescharacteristic
53C22×C7×D4C2311yescharacteristic
54C23×C4×C7—22no—
55C23×C4×C7—21no—
56C23×C4×C7—21no—
57C2×C7×D8—22no—
58C2×C7×D8—22no—
59C2×C7×D8Q814yescomplemented
60C22×C7×C8C2311yescharacteristic
61C22×C7×C8C2312yes—
62C22×C7×C8C2311yescharacteristic
63C7×C4⋊Q8—24no—
64C7×C4⋊Q8—24no—
65C7×C4⋊Q8—22no—
66C7×C4⋊Q8D414yes—
67C7×C22⋊Q8—28no—
68C7×C22⋊Q8—216no—
69C7×C22⋊Q8—44no—
70C4×C7×Q8—22no—
71C4×C7×Q8—22no—
72C4×C7×D4—24no—
73C4×C7×D4—24no—
74C4×C7×D4—22no—
75C4×C7×D4—22no—
76C4×C7×D4—22no—
77C4×C7×D4D414yes—
78C2×C7×C4⋊C4—22no—
79C2×C7×C4⋊C4—21no—
80C2×C7×C4⋊C4—21no—
81C2×C7×C4⋊C4—22no—
82C2×C7×C4⋊C4—21no—
83C2×C7×C4⋊C4D411yescharacteristic
84C2×C7×C4⋊C4D412yes—
85C2×C7×C4⋊C4C2312yes—
86C2×C7×C4⋊C4C2311yescharacteristic
87C2×C7×C4⋊C4C2312yes—
88C2×C7×C4⋊C4C2311yescharacteristic
89C2×C7×C22⋊C4—22no—
90C2×C7×C22⋊C4—21no—
91C2×C7×C22⋊C4—21no—
92C2×C7×C22⋊C4—22no—
93C2×C7×C22⋊C4—22no—
94C2×C42×C7C2312yes—
95C2×C42×C7C2311yescharacteristic
96C7×#107—24no—
97C7×#107—22no—
98C7×C4⋊C8—22no—
99C7×#102—22no—
100C7×#102—24no—
101C7×#102—24no—
102C7×#102—22no—
103C4×C7×C8—22no—
104C7×#95—22no—
105C2×D4×Q8C2×C711yescharacteristic, complemented
106C2×#474C2×C711yescharacteristic, complemented
107C2×C4×D8C2×C711yescharacteristic
108#2538C2×C711yescharacteristic, complemented
109#2057C2×C712yescomplemented
110#2023C2×C712yescomplemented
111#1990C2×C711yescharacteristic, complemented
112#1984C2×C711yescharacteristic, complemented
113#1879C2×C712yes—
114#1862C2×C712yescomplemented
115#1802C2×C711yescharacteristic, complemented
116C24×C7—21no—
117C24×C7—21no—
118C2×C7×Q8—28no—
119C2×C7×Q8—22no—
120C2×C7×Q8—24no—
121C2×C7×Q8—22no—
122C2×C7×Q8—44no—
123C2×C7×Q8—432no—
124C2×C7×Q8—44no—
125C2×C7×Q8—42no—
126C2×C7×Q8—42no—
127C2×C7×Q8D814yescomplemented
128C2×C7×D4—21no—
129C2×C7×D4—21no—
130C2×C7×D4—21no—
131C2×C7×D4—21no—
132C2×C7×D4—21no—
133C2×C7×D4—21no—
134C2×C7×D4—21no—
135C2×C7×D4—21no—
136C2×C7×D4C2×D412yes—
137C2×C7×D4C2×Q812yes—
138C2×C7×D4C2×Q812yescomplemented
139C2×C7×D4#4112yes—
140C22×C4×C7—28no—
141C22×C4×C7—22no—
142C22×C4×C7—21no—
143C22×C4×C7—24no—
144C22×C4×C7—21no—
145C22×C4×C7—22no—
146C22×C4×C7—44no—
147C22×C4×C7—44no—
148C22×C4×C7—42no—
149C22×C4×C7—42no—
150C22×C4×C7—42no—
151C22×C4×C7—42no—
152C22×C4×C7—42no—
153C22×C4×C7C2×D412yes—
154C22×C4×C7C2×D411yescharacteristic
155C22×C4×C7C2×D412yes—
156C22×C4×C7C2×D411yescharacteristic
157C22×C4×C7C2411yescharacteristic
158C7×D8—28no—
159C2×C7×C8—21no—
160C2×C7×C8—21no—
161C2×C7×C8—21no—
162C2×C7×C8—22no—
163C2×C7×C8—22no—
164C2×C7×C8—21no—
165C2×C7×C8—21no—
166C2×C7×C8—22no—
167C2×C7×C8C2×Q812yescomplemented
168C7×C4⋊C4—28no—
169C7×C4⋊C4—24no—
170C7×C4⋊C4—24no—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—22no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—21no—
176C7×C4⋊C4—22no—
177C7×C4⋊C4—22no—
178C7×C4⋊C4—42no—
179C7×C4⋊C4—44no—
180C7×C4⋊C4—81no—
181C7×C4⋊C4C2×D414yes—
182C7×C4⋊C4C2×D412yes—
183C7×C22⋊C4—28no—
184C7×C22⋊C4—24no—
185C7×C22⋊C4—42no—
186C7×C22⋊C4—44no—
187C7×C22⋊C4—82no—
188C42×C7—22no—
189C42×C7—22no—
190C42×C7C2×D412yes—
191C42×C7C2×D412yes—
192C23×Q8—21no—
193C22×D8C22×C711yescharacteristic
194D4×Q8—24no—
195D4×Q8—24no—
196C2×C4⋊Q8C22×C712yes—
197C2×C4⋊Q8C22×C711yescharacteristic
198C2×C22⋊Q8—21no—
199C2×C22⋊Q8—21no—
200C2×C22⋊Q8—22no—
201C2×C4×Q8C22×C711yescharacteristic
202C2×C4×D4C22×C712yes—
203C2×C4×D4C22×C711yescharacteristic
204C2×C4×D4C22×C711yescharacteristic
205#474—24no—
206C4×D8—24no—
207C2×#107C22×C711yescharacteristic
208C2×#107C22×C712yes—
209C2×C4⋊C8C22×C711yescharacteristic
210C2×#102C22×C712yes—
211C2×#102C22×C711yescharacteristic
212C2×#102C22×C712yes—
213C2×#102C22×C711yescharacteristic
214C2×C4×C8C22×C711yescharacteristic
215#396—22no—
216#390C22×C712yes—
217#389C22×C712yes—
218#384C22×C711yescharacteristic
219#380—21no—
220#340C22×C712yes—
221#340C22×C712yes—
222#340C22×C712yes—
223#336C22×C712yes—
224#336C22×C712yes—
225C23×C7—22no—
226C23×C7—22no—
227C23×C7—22no—
228C23×C7—22no—
229C23×C7—41no—
230C23×C7—41no—
231C23×C7—41no—
232C23×C7—41no—
233C23×C7—41no—
234C23×C7—41no—
235C23×C7—41no—
236C23×C7—41no—
237C23×C7—41no—
238C23×C7—41no—
239C23×C7C22×D411yescharacteristic
240C7×Q8—28no—
241C7×Q8—416no—
242C7×Q8—48no—
243C7×D4—22no—
244C7×D4—22no—
245C7×D4—22no—
246C7×D4—22no—
247C7×D4—22no—
248C7×D4—41no—
249C7×D4—41no—
250C7×D4C22⋊Q814yescomplemented
251C2×C4×C7—22no—
252C2×C4×C7—22no—
253C2×C4×C7—21no—
254C2×C4×C7—21no—
255C2×C4×C7—22no—
256C2×C4×C7—24no—
257C2×C4×C7—22no—
258C2×C4×C7—22no—
259C2×C4×C7—22no—
260C2×C4×C7—416no—
261C2×C4×C7—42no—
262C2×C4×C7—41no—
263C2×C4×C7—42no—
264C2×C4×C7—41no—
265C2×C4×C7—48no—
266C2×C4×C7—41no—
267C2×C4×C7—42no—
268C2×C4×C7—42no—
269C2×C4×C7—41no—
270C2×C4×C7—41no—
271C2×C4×C7—42no—
272C2×C4×C7—84no—
273C2×C4×C7#12012yes—
274C2×C4×C7#12012yes—
275C2×C4×C7C2×D814yes—
276C2×C4×C7C2×D812yes—
277C2×C4×C7AGL(1,ℤ/8)12yes—
278C2×C4×C7C22×D411yescharacteristic
279C2×C4×C7C22×D411yescharacteristic
280C2×C4×C7C22×Q811yescharacteristic
281C2×C4×C7C2×#4111yescharacteristic
282C2×C4×C72−1+411yescharacteristic
283C2×C4×C72−1+411yescharacteristic
284C7×C8—22no—
285C7×C8—41no—
286C7×C8—42no—
287C22×Q8—28no—
288C22×Q8—22no—
289C22×Q8—21no—
290C22×Q8—44no—
291C22×Q8—44no—
292C22×Q8C7×D411yescharacteristic
293C22×D4C23×C711yescharacteristic
294C22×D4C23×C711yescharacteristic
295C23×C4—22no—
296C23×C4—21no—
297C23×C4—21no—
298C2×D8—22no—
299C2×D8—22no—
300C2×D8C7×Q814yescomplemented
301C22×C8C23×C711yescharacteristic
302C22×C8C23×C712yes—
303C22×C8C23×C711yescharacteristic
304C4⋊Q8—24no—
305C4⋊Q8—24no—
306C4⋊Q8—22no—
307C4⋊Q8C7×D414yes—
308C22⋊Q8—28no—
309C22⋊Q8—216no—
310C22⋊Q8—44no—
311C4×Q8—22no—
312C4×Q8—22no—
313C4×D4—24no—
314C4×D4—24no—
315C4×D4—22no—
316C4×D4—22no—
317C4×D4—22no—
318C4×D4C7×D414yes—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4C7×D411yescharacteristic
325C2×C4⋊C4C7×D412yes—
326C2×C4⋊C4C23×C711yescharacteristic
327C2×C4⋊C4C23×C712yes—
328C2×C4⋊C4C23×C712yes—
329C2×C4⋊C4C23×C711yescharacteristic
330C2×C22⋊C4—22no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C42C23×C712yes—
336C2×C42C23×C711yescharacteristic
337#107—22no—
338#107—24no—
339C4⋊C8—22no—
340#102—22no—
341#102—24no—
342#102—22no—
343#102—24no—
344C4×C8—22no—
345#95—22no—
346C22×C7—24no—
347C22×C7—42no—
348C22×C7—42no—
349C22×C7—42no—
350C22×C7—42no—
351C22×C7—42no—
352C22×C7—42no—
353C22×C7—42no—
354C22×C7—42no—
355C22×C7—42no—
356C22×C7—81no—
357C22×C7—81no—
358C22×C7—81no—
359C22×C7—81no—
360C22×C7C2×#12011yescharacteristic
361C22×C7#54811yescharacteristic
362C22×C7D4×Q811yescharacteristic
363C22×C7C22×D811yescharacteristic
364C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
365C22×C7#57711yescharacteristic
366C22×C7#57811yescharacteristic
367C4×C7—24no—
368C4×C7—22no—
369C4×C7—44no—
370C4×C7—42no—
371C4×C7—42no—
372C4×C7—82no—
373C4×C7—81no—
374C4×C7#44714yes—
375C4×C7C2×C22⋊Q811yescharacteristic
376C4×C7#53611yescharacteristic
377C4×C7#54811yescharacteristic
378C4×C7D4×Q811yescharacteristic
379C24—21no—
380C24—21no—
381C2×Q8—28no—
382C2×Q8—22no—
383C2×Q8—24no—
384C2×Q8—22no—
385C2×Q8—44no—
386C2×Q8—432no—
387C2×Q8—44no—
388C2×Q8—42no—
389C2×Q8—42no—
390C2×Q8C7×D814yescomplemented
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4C2×C7×D412yes—
400C2×D4C2×C7×Q812yes—
401C2×D4C2×C7×Q812yescomplemented
402C2×D4C7×#4112yes—
403C22×C4—28no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4C2×C7×D412yes—
417C22×C4C2×C7×D411yescharacteristic
418C22×C4C2×C7×D412yes—
419C22×C4C2×C7×D411yescharacteristic
420C22×C4C24×C711yescharacteristic
421D8—28no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—22no—
430C2×C8C2×C7×Q812yescomplemented
431C4⋊C4—28no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—42no—
442C4⋊C4—44no—
443C4⋊C4—81no—
444C4⋊C4C2×C7×D414yes—
445C4⋊C4C2×C7×D412yes—
446C22⋊C4—28no—
447C22⋊C4—24no—
448C22⋊C4—42no—
449C22⋊C4—44no—
450C22⋊C4—82no—
451C42—22no—
452C42—22no—
453C42C2×C7×D412yes—
454C42C2×C7×D412yes—
455C2×C7—44no—
456C2×C7—82no—
457C2×C7#240911yescharacteristic
458C2×C7C2×#44711yescharacteristic
459C2×C7#301311yescharacteristic
460C2×C7#335811yescharacteristic
461C2×C7#336711yescharacteristic
462C2×C7Q8×D811yescharacteristic
463C2×C7#338411yescharacteristic
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23C22×C7×D411yescharacteristic
479Q8—28no—
480Q8—416no—
481Q8—48no—
482D4—22no—
483D4—22no—
484D4—22no—
485D4—22no—
486D4—22no—
487D4—41no—
488D4—41no—
489D4C7×C22⋊Q814yescomplemented
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—416no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—48no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—84no—
512C2×C4C7×#12012yes—
513C2×C4C7×#12012yes—
514C2×C4C2×C7×D814yes—
515C2×C4C2×C7×D812yes—
516C2×C4C7×AGL(1,ℤ/8)12yes—
517C2×C4C22×C7×D411yescharacteristic
518C2×C4C22×C7×D411yescharacteristic
519C2×C4C22×C7×Q811yescharacteristic
520C2×C4C2×C7×#4111yescharacteristic
521C2×C4C7×2−1+411yescharacteristic
522C2×C4C7×2−1+411yescharacteristic
523C8—22no—
524C8—41no—
525C8—42no—
526C7#2432611yescharacteristic, direct product factor, Sylow
527C22—24no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22C2×C7×#12011yescharacteristic
542C22C7×#54811yescharacteristic
543C22C7×D4×Q811yescharacteristic
544C22C22×C7×D811yescharacteristic
545C22C2×C7×AGL(1,ℤ/8)11yescharacteristic
546C22C7×#57711yescharacteristic
547C22C7×#57811yescharacteristic
548C4—24no—
549C4—22no—
550C4—44no—
551C4—42no—
552C4—42no—
553C4—81no—
554C4—82no—
555C4C7×#44714yes—
556C4C2×C7×C22⋊Q811yescharacteristic
557C4C7×#53611yescharacteristic
558C4C7×#54811yescharacteristic
559C4C7×D4×Q811yescharacteristic
560C2—44no—
561C2—82no—
562C2C7×#240911yescharacteristic
563C2C2×C7×#44711yescharacteristic
564C2C7×#301311yescharacteristic
565C2C7×#335811yescharacteristic
566C2C7×#336711yescharacteristic
567C2C7×Q8×D811yescharacteristic
568C2C7×#338411yescharacteristic
subgroupG/H
center, socleC23×C7C22×D4
commutator, Frattini, cosocleC22×C4C24×C7
prime radicalC22×C7×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C4×C7×Q8 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×C7×D4×Q8C2semidirect
2. C2×C7×#474C2semidirect
4. C7×#2538C2semidirect
5. C7×#2057C2semidirect
6. C7×#2023C2semidirect
7. C7×#1990C2semidirect
8. C7×#1984C2semidirect
10. C7×#1862C2semidirect
11. C7×#1802C2semidirect
45. #24326C7direct (N × H)
59. C2×C7×D8Q8semidirect
105. C2×D4×Q8C2×C7semidirect
106. C2×#474C2×C7semidirect
108. #2538C2×C7semidirect
109. #2057C2×C7semidirect
110. #2023C2×C7semidirect
111. #1990C2×C7semidirect
112. #1984C2×C7semidirect
114. #1862C2×C7semidirect
115. #1802C2×C7semidirect
127. C2×C7×Q8D8semidirect
138. C2×C7×D4C2×Q8semidirect
167. C2×C7×C8C2×Q8semidirect
250. C7×D4C22⋊Q8semidirect
300. C2×D8C7×Q8semidirect
390. C2×Q8C7×D8semidirect
401. C2×D4C2×C7×Q8semidirect
430. C2×C8C2×C7×Q8semidirect
489. D4C7×C22⋊Q8semidirect
526. C7#24326direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×C7×D4×Q82
C2×C7×#4742
C2×C4×C7×D82
C7×#25382
C7×#20572
C7×#20232
C7×#19902
C7×#19842
C7×#18792
C7×#18622
C7×#18022
#243267